Recurrence of multidimensional affine recursion in the critical case

The stochastic process defined by the affine recursion X^x_n=A_n X_{n_1}^x+b_n \in R^d, where (A_n,b_n) \in M_d \times R^d a has been widely studied during the last years due to its interest in pure and applied probability. The case the matrices A_n contract R^d in mean (i.e. their Lyapunov exponent is strictly negative) is quite well understood, but less is known in the critical situation between dilation and contraction (i.e. when the Lyapunov exponent is null). In the critical case, we investigate conditions to ensure that the processes X_n^n is recurrent (i.e it returns infinitely often in a bounded set ). In the first part of the talk I'll present the classical proof of recurrence in dimension d=1 and show what are the difficulties in higher dimension. I'll then treat the (quite easy) case when the matrices A_n are of rank one and the (not so easy case) of invertible matrices. Based on a ongoing joint work with Marc Peigné.